Question

How to simplify e^(-1/2*cos(2x))?

Answer 1

it looks simplified to me ?
@Kainui

Answer 2

Then how to simplify (e^(-sin^2 x))*e^(-1/2*cos(2x))?

Answer 3

use below exponent property :
\[\huge a^m*a^n = a^{m\color{red}{+}n}\]

Answer 4

So e^(-sin^2 x-1/2*cos(2x))?

Answer 5

yes!
\[\large e^{-\sin^2 x}*e^{-1/2*\cos(2x)} = e^{-\sin^2 x-1/2*\cos(2x)} \]

Answer 6

next, use the identity : \(\large \color{red}{\cos(2x) = 2\cos^2x - 1}\)

Answer 7

\[\large \begin{align} \\ e^{-\sin^2 x}*e^{-1/2*\cos(2x)} &= e^{-\sin^2 x-1/2*\cos(2x)} \\
&= e^{-\sin^2 x-1/2*(\color{Red}{2\cos^2x-1})} \\

\end{align} \]

Answer 8

\[\large \begin{align} \\ e^{-\sin^2 x}*e^{-1/2*\cos(2x)} &= e^{-\sin^2 x-1/2*\cos(2x)} \\
&= e^{-\sin^2 x-1/2*(\color{Red}{2\cos^2x-1})} \\
&= e^{-\sin^2 x-\color{Red}{\cos^2x+1/2}} \\
&= e^{-(\sin^2 x+\color{Red}{\cos^2x})\color{red}{+1/2}} \\

\end{align} \]

Answer 9

So the final answer is e^-1/2?

Answer 10

yes ! if your prof hates negative fractions, you may give him : \(\large \dfrac{1}{\sqrt{e}}\)

Answer 11

And how to integrate e^(-1/2) or 1/sqrt(e)?

Answer 12

\(e\) is just a number like \(1\),\(2\) or \(\pi\)

Answer 13

it is a constant.

Answer 14

\[\large \int k ~dx = kx+C\]

Answer 15

so,
\[\large \int e^{-1/2}dx = e^{-1/2} \int dx = e^{-1/2} x + C \]

Answer 16

So it's 1/sqrt(e)*x+C

Answer 17

yep !

Answer 18

But what's (1/sqrt(e)*x+C)/(e^(-1/2*cos(2x)))?

Answer 19

whats the exact+complete question you're working on ?

Answer 20

Find the general solution of y'+(2sin(x)cos(x))y=e^(-sin^2 x).

Answer 21

I see :) so you have got :
\[\large \left(ye^{-1/2\cos(2x)}\right)' = e^{-\sin^2x}*e^{-1/2\cos(2x)}\]

Answer 22

after simplifying the right hand side, you get :
\[\large \left(ye^{-1/2\cos(2x)}\right)' = e^{-1/2}\]

Answer 23

maybe I also need to convert cos(2x)=2cos^2 x-1 again

Answer 24

then you integrate both sides :

\[\large \int \left(ye^{-1/2\cos(2x)}\right)' dx = \int e^{-1/2}dx\]

Answer 25

and you arrive at :
\[\large ye^{-1/2\cos(2x)} = e^{-1/2}x+C\]

Answer 26

right ?

Answer 27

Yes, let me work it out.

Answer 28

there is a trick here to get the solution to a nice looking form

Answer 29

nvm, just simplify and see what u get

Answer 30

I got e^(-cos^2 x+1/2)y=1/sqrt(e)*x+C

Answer 31

wolfram is giving this :
http://prntscr.com/4bgh28

Answer 32

usually we leave the answer in implicit form unless somebody asks us to isolate y

Answer 33

\[\large ye^{-1/2\cos(2x)} = e^{-1/2}x+C\]

you may leave the solution like this, no need to isolate y

Answer 34

Thank you!